Method and apparatus of physical layer network coding

ABSTRACT

Provided is a physical layer network coding method and apparatus. A relay node determines reliabilities of symbols of nodes, based on a signal received from the plurality of nodes, and generates a transmission signal that maintains reliabilities of symbols that have high reliabilities and excludes components of symbols that have low reliabilities. The relay node generates the transmission signal that reduces an expected power of error, based on the received signal.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims the benefit under 35 U.S.C. §119(e) of U.S.Provisional Application No. 61/443,290, filed on Feb. 16, 2011, in theU.S. Patent and Trademark Office, and the benefit under 35 U.S.C.§119(a) of Korean Patent Application No. 10-2011-0013553 filed on Feb.16, 2011, and Korean Patent Application No. 10-2011-0042565 filed on May4, 2011, in the Korean Intellectual Property Office, the entiredisclosures of which are incorporated herein by reference.

BACKGROUND

1. Field

The following description relates to a physical layer network codingmethod and apparatus for correcting errors that are generated in anetwork.

2. Description of Related Art

Information that is transmitted over a communication network may becoded. For example, a network coding scheme is a scheme in which codingof information is performed in intermediate nodes, as well as in asource and at a destination. A general communication network transfers,to the destination, a payload of a packet that is generated at thesource without changing the payload of the packet at a router.Conversely, a communication network using the network coding scheme mayallow changing of the payload, such as changing by mixing differentpackets at a router.

Originally, the network coding scheme was suggested to improve amulti-cast throughput in a wired network. Currently, the use of anetwork coding scheme that provides tangible effects in a wirelessnetwork has drawn attention.

SUMMARY

In one general aspect, there is provided a communication method of arelay node, the method including receiving a signal including a firstsymbol through a k^(th) symbol that are transmitted from a first nodethrough a k^(th) node, respectively, calculating, based on apredetermined criterion, a reliability of the first symbol through thek^(th) symbol, respectively, selecting one or more symbols from amongthe first through the k^(th) symbol that have a reliability that isgreater than or equal to the predetermined criterion, and generating atransmission signal that maintains the reliabilities of the selectedsymbols and that excludes components of symbols that have reliabilitieswhich are less than the predetermined criterion.

The generating may comprise generating the transmission signal todecrease an expected power of error between the transmission signal andthe received signal.

The calculating may comprise calculating the reliabilities of the firstsymbol through the k^(th) symbol, based on a log likelihood ratio (LLR)with respect to the first symbol through the k^(th) symbol,respectively, each LLR calculated based on the received signal andchannel information that is associated with the first node through thek^(th) node, respectively.

The generating may comprise generating the transmission signal in whichLLRs of the selected symbols are equivalent to LLRs of the selectedsymbols in the received signal or are within a predetermined range.

The equivalence or the difference in the predetermined range may bedetermined based on the Kuliback-Leibler distance.

The method may further comprise estimating channels with respect to thefirst node through the k^(th) node based on pilots that are transmittedfrom the first node through the k^(th) node, respectively.

The method may further comprise transmitting the transmission signal toa first destination node through a k^(th) destination node correspondingto the first node through the k^(th) node, respectively.

The transmitting may comprise transmitting the transmission signal byscaling the transmission signal, based on a predetermined transmissionpower.

The method may further comprise transmitting identification informationthat is associated with nodes corresponding to the selected symbols.

In another aspect, there is provided a computer-readable storage mediumhaving stored therein program instructions to cause a processor toimplement a communication method of a relay node, the method includingreceiving a signal including a first symbol through a k^(th) symbol thatare transmitted from a first node through a k^(th) node, respectively,calculating, based on a predetermined criterion, a reliability of thefirst symbol through the k^(th) symbol, respectively, selecting one ormore symbols from among the first through the k^(th) symbol that have areliability that is greater than or equal to the predeterminedcriterion, and generating a transmission signal that maintains thereliabilities of the selected symbols and that excludes components ofsymbols that have reliabilities which are less than the predeterminedcriterion.

In another aspect, there is provided a relay node including a receivingunit to receive a signal including a first symbol through a k^(th)symbol that are transmitted from a first node through a k^(th) node,respectively, and a processing unit to calculate, based on apredetermined criterion, reliabilities of the first symbol through thek^(th) symbol, respectively, wherein the processing unit selects one ormore symbols that have a reliability that is greater than or equal tothe predetermined criterion from among the first symbol through thek^(th) symbol, and generates a transmission signal that maintains thereliabilities of the selected symbols and that excludes components ofsymbols that have reliabilities which are less than the predeterminedcriterion.

The processing unit may generate the transmission signal to decrease anexpected power of error between the transmission signal and the receivedsignal.

The processing unit may calculate the reliabilities of the first symbolthrough the k^(th) symbol, based on log likelihood ratio (LLR) withrespect to the first symbol through the k^(th) symbol, respectively,each LLR calculated based on the received signal and channel informationthat is associated with the first node through the k^(th) node,respectively.

The processing unit may generate the transmission signal in which LLRsof the selected symbols are equivalent to LLRs of the selected symbolsin the received signal or are within a predetermined range.

The equivalence or the difference in the predetermined range may bedetermined based on the Kullback-Leibler distance.

The processing unit may estimate channels with respect to the first nodethrough the k^(th) node based on pilots that are transmitted from thefirst node through the k^(th) node, respectively.

The relay node may further comprise a transmitting unit to transmit thetransmission signal to a first destination node through a k^(th)destination node corresponding to the first node through the k^(th)node, respectively.

The transmitting unit may scale the transmission signal based on apredetermined transmission power, and transmit the scaled transmissionsignal.

The transmitting unit may transmit identification information that isassociated with nodes corresponding to the selected symbols.

In another aspect, there is provided a relay node in a wireless network,the relay node including a receiver configured to receive symbols fromone or more nodes that are within the wireless network, a processorconfigured to determine a reliability of each received symbol based on aphysical layer network coding method that uses reliability, and atransmitter configured to transmit only those received symbols that aredetermined to have a reliability above a threshold.

The physical layer network coding method may calculate the reliabilityof a received symbol based on a log likelihood ratio (LLR) of thereceived symbol and channel information that is associated with a nodethat transmitted the received symbol.

The receiver may be further configured to simultaneously receive a firstsymbol from a first node and a second symbol from a second node, and theprocessor may be further configured to determine a reliability of thefirst symbol and the second symbol based on the physical layer networkcoding method that uses reliability.

In response to the receiver determining that the first symbol has areliability above the threshold, and that the second symbol has areliability below the threshold, the transmitter may be furtherconfigured to transmit a transmission signal including the first symboland excluding the second symbol.

Other features and aspects may be apparent from the following detaileddescription, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating an example of an algebraic networkcoding scheme.

FIG. 2 is a diagram illustrating an example of an analog network codingscheme.

FIG. 3 is a diagram illustrating an example of a physical layer networkcoding method that uses reliability.

FIG. 4 is a diagram illustrating an example of a network to which aphysical layer network coding method is applied.

FIGS. 5 through 7 are graphs illustrating examples of a result of asimulation based on a physical layer network coding method that usesreliability.

FIG. 8 is a flowchart illustrating an example of a communication methodof a relay node based on a physical layer network coding method thatuses reliability.

FIG. 9 is a diagram illustrating an example of a relay node that uses aphysical layer network coding method that uses reliability.

Throughout the drawings and the detailed description, unless otherwisedescribed, the same drawing reference numerals should be understood torefer to the same elements, features, and structures. The relative sizeand depiction of these elements may be exaggerated for clarity,illustration, and convenience.

DETAILED DESCRIPTION

The following detailed description is provided to assist the reader ingaining a comprehensive understanding of the methods, apparatuses,and/or systems described herein. Accordingly, various changes,modifications, and equivalents of the methods, apparatuses, and/orsystems described herein may be suggested to those of ordinary skill inthe art. Also, descriptions of well-known functions and constructionsmay be omitted for increased clarity and conciseness.

A network error correcting method based on a physical layer networkcoding method may be used to correct errors that occur betweencommunication links when a network coding scheme is used in a wirelessad hoc network environment. The network error correcting method thatuses the physical layer network coding method may be applied without asignificant change regardless of whether a number of nodes increases ordecreases, and may be applicable to a wireless sensor networkenvironment.

FIG. 1 illustrates an example of an algebraic network coding scheme.

Referring to FIG. 1, node V₁ is to transmit data X₁ to node t₁, and nodeV₂ is to transmit data X₂ to node t₂. In this example, node t₁ islocated outside a coverage area 110 of node V₁, and node t₂ is locatedoutside a coverage area 120 of node V₂. Accordingly, node V₁ and node V₂may transmit data to node t₁ and node t₂, respectively, via a relay nodeV₃.

The algebraic network coding scheme may be a packet level network codingscheme. For example, node V₁ and node V₂ may transmit signal X₁(w₁) andsignal X₂(w₂) to node V₃, respectively. In this example, node V₁ andnode V₂ may stagger the transmission of signal X₁ and signal X₂. Relaynode V₃ may decode a received signal to obtain data {tilde over (w)}₁and data {tilde over (w)}₂, and may apply the network coding scheme todata {tilde over (w)}₁ and data {tilde over (w)}₂ to generate atransmission signal {tilde over (w)}₁□{tilde over (w)}₂. Relay node V₃may broadcast the transmission signal.

Node t₁ and node t₂ are located inside a coverage area 130 of node V₃and may receive the transmission signal of the relay node V₃.Accordingly, node t1 may obtain signal X₁ by network decoding signal X₂that is received from node V₂ and the transmission signal from relaynode V₃. In the same manner, node t₂ may obtain data X₂ by networkdecoding signal X₁ that is received from node V₁ and the transmissionsignal from relay node V₃.

In the algebraic network coding scheme, a relay node decodes each ofsignals that are received from transmission nodes, and thus, the relaynode has a high complexity. Also, the algebraic network coding schememay be vulnerable to interference from other transmission nodes.

FIG. 2 illustrates an example of an analog network coding scheme.

Referring to FIG. 2, similar to FIG. 1, node V₁ is to transmit signal X₁to node t₁, and node V₂ is to transmit signal X₂ to node t₂. In thisexample, node V₁ and node V₂ transmit data to node t₁ and node t₂, viarelay node V₃.

In the analog network coding scheme, node V₁ may transmit signal X₁ torelay node V₃, and simultaneously, node V₂ may transmit signal X₂ torelay node V₃. Relay node V₃ may amplify a received signal includingsignal X₁ and signal X₂ to broadcast a generated transmission signal,without decoding the received signal.

Node t₁ may obtain signal X₁ based on signal X₂ that is received fromnode V₂ and the transmission signal from relay node V₃, and node t₂ mayobtain signal X₂ based on signal X₁ that is received from node V₁ andthe transmission signal from relay node V₃.

In the analog network coding scheme, a relay node may merely amplify andtransmit a received signal, which also amplifies noise. Accordingly, adestination node may be significantly affected by noise.

Therefore, there is a need for a physical layer network coding methodthat is robust and scalable to overcome a drawback of the conventionalnetwork coding scheme. In various examples herein, a relay node maygenerate a transmission signal by removing a component that has a lowreliability from a received signal, and may broadcast the generatedtransmission signal.

Various scenarios are provided as examples in the following. Forexample, node V_(j) receives packets P_(1,j), P_(2,j), . . . , P_(k,j)from transmission nodes V_(i) (i=1, 2, . . . , k). Each transmissionnode V_(i) may use a transmission power P. In this example, the packetP_(i,j) is encoded as l constellation symbols X_(i,j) ¹X_(i,j) ² . . .X_(i,j) ^(l). A signal received by node V_(j) Y_(j) ¹Y_(j) ² . . . Y_(j)^(l).

A received packet Y_(j) ^(m) (m=1, 2, . . . , l) may be modeled as acorrupted version of a scaled versions of X_(i,j) ^(m) by n_(j) ^(m)that are independent identically distributed (i.i.d.) samples ofadditive noise n_(j) that have a mean of zero and a variance of σ² percomplex dimension. Y_(j) ^(m) may be expressed by equation

$Y_{j}^{m} = {{\sum\limits_{i = 1}^{k}{\alpha_{i,j}X_{i,j}^{m}}} + n_{j}^{m}}$

(m=1, 2, . . . , and l).

In this equation, α_(i,j) denotes a channel coefficient of channelbetween node V_(i) to relay node V_(j).

In the conventional network coding scheme, symbols X_(i,j) ¹X_(i,j) ² .. . X_(i,j) ^(l) (i=1, 2, . . . , k) may be selected from the samelattice L. The selected lattice may be an integer lattice Z¹ or may beanother lattice that has higher shaping gains. Relay node V_(j) maycalculate {tilde over (Y)}_(j) ^(m)=Σ_(i=1) ^(k)b_(i)X_(i,j) ^(m) (m=1,2, . . . , l) and a coefficient (b₁, . . . , b_(k)), based on thereceived packet Y_(j) ^(m). In this example, the coefficient (b₁, . . ., b_(k)) may be an integer vector. Relay node V_(j) may transmit atransmission signal {tilde over (Y)}_(j) ¹{tilde over (Y)}_(j) ² . . .{tilde over (Y)}_(j) ^(l).

To generate the transmission signal {tilde over (Y)}_(j) ¹{tilde over(Y)}_(j) ² . . . {tilde over (Y)}_(j) ^(l), relay node V_(j) may selecta factor λ. The factor λ may be a factor that enables (λα_(1,j),λα_(2,jj, . . . , λα) _(k,j)) to be as close to a point (b₁, . . . ,b_(k)) of the integer lattice Z^(k). If the point (b₁, . . . , b_(k)) isa point closest to the integer lattice Z^(k) to (λα_(1,j), λα_(2,jj), .. . , λα_(k,j)), a signal-to-noise ratio (SNR) of the transmissionsignal of relay node V_(j) may be represented by

${S\; N\; R} = {\frac{P}{\lambda^{2} + {\sum\limits_{i = 1}^{k}{P{{{\lambda\alpha}_{i,j} - b_{j}}}^{2}}}}.}$

Therefore, the factor λ that increases the SNR the most may bedetermined.

The conventional schemes assume that receiver nodes are informed of achannel coefficient α_(i,j). However, if a relay node V_(j) is notcompletely aware of the channel coefficient α_(i,j) and a high α_(i,j)is requested, an actual SNR of the transmission signal {tilde over(Y)}_(j) ¹{tilde over (Y)}_(j) ² . . . {tilde over (Y)}_(j) ^(l) may belowered. According to the conventional schemes, performance may be basedon α_(i,j) when the receiver nodes are not completely aware of thechannel coefficient α_(i,j).

Therefore, a robust physical layer network coding method may berequested. A wireless sensor network that has a large change in topologybased on access of member nodes may have robustness, scalability, andlocality. Therefore, there is a need for a physical layer network codingmethod in which each node does not need a large amount of informationassociated with a network topology, an algorithm does not need to besignificantly changed based on a change in a number of nodes, and inwhich performance is high.

Various examples herein are directed towards a physical layer networkcoding method that enhances the robustness by removing components ofsignals that have low reliabilities at each node. Hereinafter, anexample of the physical layer network coding method is described withreference to a system model of FIG. 3.

FIG. 3 illustrates an example of a physical layer network coding methodthat uses reliability.

Referring to FIG. 3, node V₁ transmits a bit b₁ based on a binary phaseshift keying (BPSK) modulation scheme, and simultaneously, node V₂transmits a bit b₂ based on the BPSK modulation scheme. In this example,node V₃ corresponds to a relay node. Node V₃ may simultaneously receivesignals that are transmitted from node V₁ and node V₂. In this example,it is assumed that a channel between nodes is a complex Gaussianchannel. A channel coefficient of a channel between node V₁ and node V₃,and a channel coefficient of a channel between node V₂ and node V₃ areα_(1,3) and α_(2,3), respectively. The received signal r at node V₃ maybe modeled as shown below:

r=α _(1,3) x ₁+α_(2,3) x ₂ +n

In this example, x_(i)(−1)^(b) ^(i) for i=1,2 and n denotes noise. Inthis example, n has a mean of zero and a variance of σ²/2 per realdimension.

In this example, a log likelihood ratio (b₁) (LLR (b₁)) is greater thana predetermined threshold and an LLR (b₂) is not. Here, LLR (b)=ln{p(b=01r)/p(b=1|r)}. Also in this example, the bit b₁ transmitted fromnode V₁ is greater than a predetermined criterion, and the bit b₂transmitted from node V₂ is less than the predetermined criterion.

Node V₃ may generate a transmission signal ({tilde over (r)}) asexpressed by

{tilde over (r)}=α _(1,3) x ₁ +ñ

In this example, ñ is an additive white Gaussian noise (AWGN) that has amean of zero and a variance of σ² per real dimension. Also, σ² is avalue to be optimized.

In this example, {tilde over (r)} may be generated to have adistribution of an LLR that is approximately the same as the LLR (b₁),based on the Kullback-Leibler distance. Also, {tilde over (r)} may begenerated to include minimum noise. Accordingly, {tilde over (r)} may begenerated to minimize or otherwise reduce an expected power of error.{tilde over (r)} may be generated to decrease the expected power oferror and to remove an effect of the bit b₂ of which a reliability islowered by Gaussian noise.

The physical layer network coding method may consume power fortransmitting a bit that has a high reliability, as opposed totransmitting a bit that has a low reliability, and thus, may improveperformance in transmission.

Hereinafter, an example of a method of node V₃ that generates {tildeover (r)} is described. If a signal transmitted from node V_(i) has ahigh reliability, and a signal transmitted from node V₂ has a lowreliability, node V₃ may generate {tilde over (r)} as given inEquation 1. As another example, if the signal from node V₁ has a lowreliability and a signal from node V₂ has a high reliability, thetransmission signal {tilde over (r)} generated by node V₃ may beα_(2,3)x+ñ. If both the signal from node V₁ and the signal from node V₂have high reliabilities, {tilde over (r)} generated by node V₃ may beα_(1,3)x+α_(2,3)x+ñ. If both the signal from node V₁ and the signal fromnode V₂ have low reliabilities, node V₃ may not generate {tilde over(r)}.

{tilde over (r)}=α _(1,3) x+ñ  [Equation 1]

In Equation 1, x is selected from signal constellation A1={c1, c2, . . ., cr1} and ñ is an AWGN having a mean of zero and a variance of σ² percomplex dimension.

Assuming that p_(i) (=p(x_(i)=c_(i)|r))

p(x=c_(i)|{tilde over (r)}) with respect to all i=1, 2, . . . , r₁, anexpected power of error may be expressed by Equation 2.

$\begin{matrix}{\sum\limits_{i = 1}^{r_{i}}{p_{i}{{\overset{\sim}{r} - c_{i}}}^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

Node V₃ may generate {tilde over (r)} that minimizes the expected powerof error.

For example, assuming that d_(i)=|{tilde over (r)}−c_(i)|, i=1, 2, . . ., r₁, Equation 3 may be obtained without loss of generality byre-labeling.

p ₁ ≧p ₂ ≧p ₃ ≧ . . . ≧p _(r) ₁   [Equation 3]

Equation 4 may be obtained by assuming that noise ñ is the Gaussianmodel.

$\begin{matrix}{\frac{p_{1}}{p_{i}} = \frac{\exp\left( {- \frac{d_{1}^{2}}{\sigma_{2}^{2}}} \right)}{\exp\left( {- \frac{d_{i}^{2}}{\sigma_{2}^{2}}} \right)}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

(for i=1, 2, . . . , r₁)

In this example,

${{\ln \left( p_{1} \right)} - {\ln \left( p_{i} \right)} + \frac{d_{1}^{2}}{\sigma_{2}^{2}}} = \frac{d_{i}^{2}}{\sigma_{2}^{2}}$

may be obtained by modifying Equation 4, and Equation 5 may be obtainedby modifying Equation 2 that is associated with the expected power oferror.

$\begin{matrix}{{\frac{1}{\sigma_{2}^{2}}{\sum\limits_{i = 1}^{r_{i}}{p_{i}{{\overset{\sim}{r} - c_{i}}}^{2}}}} = {\frac{d_{i}^{2}}{\sigma_{2}^{2}} + {\log \left( p_{1} \right)} - {\sum\limits_{i = 1}^{r_{1}}{p_{i}{\ln \left( p_{i} \right)}}}}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

Assuming that H(P)=−Σ_(i=1) ^(r) ¹ p_(i) ln(p_(i)) is a natural entropyof a distribution p_(i), Equation 5 may be modified to Equation 6.

$\begin{matrix}{{\frac{1}{\sigma_{2}^{2}}{\sum\limits_{i = 1}^{r_{i}}{p_{i}{{\overset{\sim}{r} - c_{i}}}^{2}}}} = {\frac{d_{i}^{2}}{\sigma_{2}^{2}} + {\log \left( p_{1} \right)} + {H()}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

In this example, if p_(j) is substituted for p₁, Equation 7 may beobtained with respect to an optimal {tilde over (r)}.

$\begin{matrix}{{\frac{d_{j}^{2}}{\sigma_{2}^{2}} + {\log \left( p_{j} \right)} + {H()}} = {K_{o}\mspace{14mu} \left( {{{{for}\mspace{14mu} {all}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},r_{1}} \right)}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

In Equation 7, K_(o) denotes a constant. Accordingly, K_(o) and σ² thatminimize Equation 8 representing the expected power may be selected.

$\begin{matrix}{{\sum\limits_{i = 1}^{r_{i}}{p_{i}{{r - c_{i}}}^{2}}} = {d_{1}^{2} + {\sigma_{2}^{2}\left( {{\ln \left( p_{1} \right)} + {H()}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

For simplicity, assuming that

${a_{j} = {{\ln \left( p_{j} \right)} + {H()}}},{{\frac{d_{j}^{2}}{\sigma_{2}^{2}} + a_{j}} = {K_{o}.}}$

<Scheme for a Constellation Having Two Elements>

A method of calculating an optimal {tilde over (r)} when signalconstellation A1 has two elements, namely, when r₁=2, is described.

First, an objective function such as the expected power of error d₁ ²+σ₂²(ln(p_(l))+H(P)), may be minimized with respect to a fixed σ2 bydecreasing d₁. When d₁ decreases based on Equation 7, d₂ may alsodecrease. However, d₁+d₂≧|c₁−c₂|, based on a triangle inequality.

Therefore, d₁ and d₂ that minimize the expected power of error withrespect to the fixed σ2 may be obtained by Equation 9.

d ₁ d ₂ =|c ₁ −c ₂|  [Equation 9]

When d=|c₁−c₂|, Equation 10 may be obtained from

${\frac{d_{j}^{2}}{\sigma_{2}^{2}} + a_{j}} = {K_{o}.}$d ₂ ² −d ₁ ²=σ₂ ²(a ₁ −a ₂)  [Equation 10]

Therefore, a difference between d₂ and d₁ may be given by Equation 11.

$\begin{matrix}{{d_{2} - d_{1}} = {\frac{\sigma_{2}^{2}\left( {a_{1} - a_{2}} \right)}{d_{1} + d_{2}} = \frac{\sigma_{2}^{2}\left( {a_{1} - a_{2}} \right)}{d}}} & \left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack\end{matrix}$

d₂ and d₁ may be expressed by Equation 12.

$\begin{matrix}{{d_{1} = {\frac{d}{2} - \frac{\sigma_{2}^{2}\left( {a_{1} - a_{2}} \right)}{2d}}},{d_{2} = {\frac{d}{2} + {\frac{\sigma_{2}^{2}\left( {a_{1} - a_{2}} \right)}{2d}.}}}} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

The expected power of error may be expressed by Equation 13, based onEquation 12.

$\begin{matrix}{{d_{1}^{2} + {a_{1}\sigma_{2}^{2}}} = {\left( {\frac{d}{2} - \frac{\sigma_{2}^{2}\left( {a_{1} - a_{2}} \right)}{2d}} \right)^{2} + {a_{1}\sigma_{2}^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

When Equation 13 is differentiated with respect to σ₂ ², σ₂ ² thatminimizes the expected power of error may be obtained as expressed byEquation 14.

$\begin{matrix}{\sigma_{2}^{2} = {{- \frac{a_{1} + a_{2}}{\left( {a_{1} - a_{2}} \right)^{2}}}d^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack\end{matrix}$

In this example, d₁ and d₂ may be expressed by Equation 15, based onEquation 14.

$\begin{matrix}{{d_{1} = {\frac{a_{1}}{a_{1} - a_{2}}d}},{d_{2} = {\frac{- a_{2}}{a_{1} - a_{2}}{d.}}}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

Equation 15 may be simplified as given in Equation 16.

d ₁ =p ₂ d

d ₂ =p ₁ d  [Equation 16]

The optimal {tilde over (r)} may be expressed by Equation 17.

{tilde over (r)}=p ₁ c ₁ +p ₂ c ₂  [Equation 17]

<Scheme for a Constellation Having Two or More Elements>

For a given p_(i)(=p(xi=ci|r)), i=1, 2, . . . , r₁, a value for {tildeover (r)} that always satisfies Equation 18 may not exist.

p _(i) =p(x=c _(i) |{tilde over (r)}), i=1,2, . . . ,r ₁  [Equation 18]

When {tilde over (r)} that satisfies Equation 18 exist, Equation 19 maybe satisfied for all i≠j.

$\begin{matrix}{{{\ln \left( p_{i} \right)} - {\ln \left( p_{j} \right)}} = {{- \frac{d_{j}^{2}}{\sigma_{2}^{2}}} + \frac{d_{i}^{2}}{\sigma_{2}^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$

The equations in Equation 19 may correspond to lines that areperpendicular to a line segment between c_(i) and c_(j).

The lines may not always meet at the same {tilde over (r)}. If the linesmeet at the same {tilde over (r)}, p_(i) (i=1, 2, . . . , r1) may be ageometrically consistent probability distribution function (PDF).

In this example, complete matching may be difficult, and thus, there isa need for an {tilde over (r)} that enables an a posteriori PDF, thatis, p(x=ci|{tilde over (r)}) to be as close to p_(i) based on theKullback-Leibler distance, and that minimizes the expected power oferror corresponding to Σ_(i=1) ^(T) ^(i) P_(i)|{tilde over (r)}−c_(i)|².

The a posteriori PDF induced by {tilde over (r)} may be expressed byEquation 20.

$\begin{matrix}{{\left( c_{i} \right)} = {{p\left( c_{i} \middle| \overset{\sim}{r} \right)} = \frac{\exp\left( {- \frac{d_{i}^{2}}{\sigma_{2}^{2}}} \right)}{\sum\limits_{j = 1}^{r_{i}}\; {\exp\left( {- \frac{d_{j}^{2}}{\sigma_{2}^{2}}} \right)}}}} & \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

The Kullback-Leibler distance between a distribution P and an aposteriori distribution may be expressed by Equation 21.

$\begin{matrix}{D\left( {{\left.  \right)} = {{- {H()}} + {\frac{1}{\sigma_{2}^{2}}{\sum\limits_{j = 1}^{r_{i}}\; {p_{j}d_{j}^{2}}}} + {\ln\left( {\sum\limits_{j = 1}^{r_{i}}\; {\exp\left( {- \frac{d_{j}^{2}}{\sigma_{2}^{2}}} \right)}} \right)}}} \right.} & \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack\end{matrix}$

In this example, Σ_(j=1) ^(r) ^(i) p_(j)d_(j) ² is the expected power oferror. At a high SNR, the approximation may be based on Equation 22.

$\begin{matrix}{{{\ln\left( {\sum\limits_{j = 1}^{r_{i}}\; {\exp\left( {- \frac{d_{j}^{2}}{\sigma_{2}^{2}}} \right)}} \right)} \simeq \frac{- {\min_{j}\left( d_{j}^{2} \right)}}{\sigma_{2}^{2}}} = \frac{- d_{1}^{2}}{\sigma_{2}^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

Therefore, Equation 21 may be modified as Equation 23.

$\begin{matrix}{D\left( {{\left.  \right)} \simeq {{- {H()}} + {\frac{1}{\sigma_{2}^{2}}{\left\{ {\left( {\sum\limits_{j = 1}^{r_{i}}\; {p_{j}d_{j}^{2}}} \right) - d_{1}^{2}} \right\}.}}}} \right.} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

At a high SNR, a correct {tilde over (r)} and a most adjacent point tothe correct {tilde over (r)} may have non-negligible posteriorprobabilities p₁ and p₂. In this example, d₁ ²≅p₂ ²d₂ as describedherein, and thus, the minimization of the Kullback-Leibler distance maybe equivalent to the minimization of the expected power of errorcorresponding to Σ_(j=1) ^(r) ¹ p_(j)d_(j) ². Therefore, {tilde over(r)} may be determined as expressed by Equation 24.

$\begin{matrix}{\overset{\sim}{r} = {\sum\limits_{i = 1}^{r_{1}}\; {p_{i}c_{i}}}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack\end{matrix}$

Various examples are further provided herein. A first example describesBPSK transmission by two source nodes V₁=S₁ and V₂=S₂, and reception byan intermediate node V₃. A second example describes a case in which ahigher order constellation and three or more transmission nodes exist. Athird example describes transmission from arbitrary nodes, for example,relay nodes or source nodes, to arbitrary nodes.

<BPSK Transmission Example>

Node V₃ corresponding to a relay node receives source packets P_(1,3)and P_(2,3) from node V_(i) (i=1,2) at an average transmission power Pper node. In this example, packet Pi,j is encoded as 1 constellationsymbols as shown in Equation 25.

X _(i,3) ¹ X _(i,3) ² . . . X _(i,3) ^(l)  [Equation 25]

A signal received by node V₃ is expressed by Equation 26.

Y ₃ ¹ Y ₃ ² . . . Y ₃ ^(l)  [Equation 26]

The received signal (Y₃ ^(m)) (m=1, 2, . . . , |) may be modeled as acorrupted version of a scaled version of X_(i,j) ^(m) by n₃ ^(m). Inthis example, n₃ ^(m) is an i.i.d. sample of complex additive noise thathas a mean of zero and a variance of σ²/2 per real dimension. Y₃ ^(m)may be expressed by Equation 27.

$\begin{matrix}{Y_{3}^{m} = {{\sum\limits_{i = 1}^{2}\; {\alpha_{i,3}X_{i,3}^{m}}} + n_{3}^{m}}} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack\end{matrix}$

(for m=1, 2, . . . , l)

In Equation 27, α_(i,3) is a channel coefficient of a channel from nodeV_(i) (i=1, 2) to the node V₃.

In various examples, a potentially coded signal may be used, and thus,node V₃ may perform symbol-by-symbol transmission to reduce a complexityof node V₃. Node V₃ may regard BPSK symbols X_(i,3) ¹X_(i,3) ² . . .X_(i,3) ^(l) (i=1, 2) as uncoded symbols and may calculate LLRs ofsymbols X_(i,3) ^(m) (i=1, 2) from Y_(j) ^(m) (m=1, 2, . . . , l). NodeV₃ may also calculate an average function of the LLRs as expressed byEquation 28.

$\begin{matrix}{{LLR}_{i} = {{L\; L\; {R\left( {X_{i,3}^{1}X_{i,3}^{2}\mspace{14mu} \ldots \mspace{14mu} X_{i,3}^{1}} \right)}} = \frac{\sum\limits_{m = 1}^{1}{{L\; L\; {R\left( X_{i,3}^{m} \right)}}}}{1}}} & \left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

Other functions may be applicable. However, an LLR that is obtained fromtransmission of a BPSK uncoded signal may show that the LLR may beclosely related to an average virtual SNR of each symbol stream. Node V₃may select thresholds (T₃ ¹ and T₃ ²), in advance. In this example, T₃ ¹and T₃ ² denote LLR qualities of data streams, appropriated fordecoding, which are from node V₁ and node V₂, respectively.

On the assumptions set forth in the forgoing, three cases may bepossible based on a reliability of Y₃ ^(m) of node V₃.

-   -   Case 1: LLR₁>T₃ ¹ and LLR₂<T₃ ² or LLR₁<T₃ ¹ and LLR₂>T₃ ²    -   Case 2: all LLR_(i)>T₃ ¹ for i=1, 2    -   Case 3: all LLR_(i)≦T₃ ^(i) for i=1, 2    -   Case 1 describes an example in which LLR₁>T₃ ¹ and LLR₂<T₃ ²        without loss of generality.

In this example, node V₃ may generate a transmission signal ({tilde over(Y)}₃ ^(m)) (m=1, 2, . . . , l) that has that p(X_(1,3) ^(m)|{tilde over(Y)}₃ ^(m)) has the same a posteriori probability distribution asp(X_(1,3) ^(m)|Y₃ ^(m)), and that minimizes an expected power of error.A method of generating {tilde over (Y)}₃ ^(m) has been previouslydescribed and is given in Equation 29.

{tilde over (Y)} ₃ ^(m)=α_(1,3) X _(1,3) ^(m) +ñ ₃ ^(m)  [Equation 29]

In this example, {tilde over (Y)}₃ ^(m) may have the same LLR as LLR₁with respect to X_(1,3m), and thus, from a point of view of receivernode t₁, {tilde over (Y)}₃ ^(m) may be equivalent to X_(1,3) ^(m) and{tilde over (Y)}₃ ^(m) may not carry information that is associated withX_(2,3) ^(m). Node V₃ may scale {tilde over (Y)}₃ ¹{tilde over (Y)}₃ ² .. . {tilde over (Y)}₃ ^(l) using a constant factor β so that a sequence(β{tilde over (Y)}₃ ^(m)) (m=1, 2, . . . , l) may have an average powerP. Node V₃ may perform scaling to satisfy another predetermined powerconstraint, such as a peak power. Node V₃ may transmit β{tilde over(Y)}₃ ^(m). In this example, the information that is associated withX_(2,3) ^(m) may be removed from a header of the transmitted sequence.That is, identification information associated with node V₁corresponding to X_(1,3) ^(m) may be included in a header of β{tildeover (Y)}₃ ^(m). Accordingly, information that is associated withtransmission nodes corresponding to symbols that have high reliabilitiesmay be included in a header of a sequence.

Case 2 describes an example in which X_(i,3) ¹X_(i,3) ² . . . X_(i,3)^(l), i=1,2 have high reliabilities, respectively. Node V₃ may generatea transmission signal that satisfies a condition of Case 2 and minimizesan expected power of error. In this example, a set of four possiblevalues of Σ_(i=1) ²α_(i,3)X_(i,3) ^(m), given by A={±α_(1,3)±α_(2,3)},may be used.

Here, Y₃ ^(m) of node V₃ may be expressed by Equation 30.

Y ₃ ^(m) =X ₃ ^(m) +n ₃ ^(m) (m=1,2, . . . ,l)  [Equation 30]

Also, X₃ ^(m)=α_(1,3)X_(1,3) ^(m)+α_(2,3)X_(2,3) ^(m)εA. Node V₃ maycalculate {tilde over (Y)}₃ ^(m) that has p(X₃ ^(m)|{tilde over (Y)}₃^(m)) that has the same posterior distribution as p(X₃ ^(m)|Y₃ ^(m))based on the Kullback-Leibler distance, and that minimizes an expectedpower of error with respect to all m=1, 2, . . . , l.

{tilde over (Y)} ₃ ^(m) =X ₃ ^(m) +ñ ₃ ^(m)  [Equation 31]

The method of generating {tilde over (Y)}₃ ^(m) has previously beendescribed herein. Node V₃ may scale {tilde over (Y)}₃ ¹{tilde over (Y)}₃² . . . {tilde over (Y)}₃ ^(l) using a constant factor β so that β{tildeover (Y)}₃ ^(m) (m=1, 2, . . . , l) may have an average power P. Also,the scaling may be performed to satisfy another predetermined powerconstraint, such as a peak power. Node V₃ may transmit β{tilde over(Y)}₃ ^(m). That is, information that is associated with transmissionnodes corresponding to symbols that have high reliabilities may beincluded in a sequence header. Accordingly, information that isassociated with all the nodes in Case 2 may be included in a header ofβ{tilde over (Y)}₃ ^(m).

Case 3 describes an example in which X_(i,3) ¹X_(i,3) ² . . . X_(i,3)^(l), i=1,2 have reliabilities that are less than or equal to apredetermined criterion, respectively. Accordingly, node V₃ may nottransmit a stream that is associated with X_(i,3) ¹X_(i,3) ² . . .X_(i,3) ^(l), i=1,2. Any further amplification or re-transmission maydecrease a quality of an underlying stream in a transmission signal.Accordingly, node V₃ may cease transmission and may consume energy.

<Transmission Method for a Higher Order Constellation>

The descriptions herein may be extended to a case in which underlyingsignals use a higher order constellation, for example, quadraturephase-shift keying (QPSK), 8-phase-shift keying (PSK), and 16-QuadratureAmplitude Modulation (16-QAM). In this example, signal constellations innode V₁=S₁ and node V₂=S₂ are referred to as A₁ and A₂, respectively. Areceived signal (Y₃ ^(m)) of node V₃ may be expressed by Equation 32.

$\begin{matrix}{Y_{3}^{m} = {{\sum\limits_{i = 1}^{2}\; {\alpha_{i,3}X_{i,3}^{m}}} + n_{3}^{m}}} & \left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack\end{matrix}$

(m=1, 2, . . . , l)

In Equation 32, α_(i,3) is a channel coefficient of a channel betweennode V_(i) (i=1, 2) and node V₃.

For each of i=1, 2, m=1, 2, . . . , l and cεA_(i), an LLR may becalculated as expressed by Equation 33.

$\begin{matrix}{{{LLR}_{i,m}(c)} = {\ln \frac{p\left( {X_{i,3}^{m} = \left. c \middle| r \right.} \right)}{p\left( {X_{i,3}^{m} \neq c} \middle| r \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack\end{matrix}$

In this example,

${L\; L\; {R\left( X_{i,3}^{m} \right)}} = {\max\limits_{c \in _{i}}{{LLR}_{i,m}(c)}}$

(i=1, 2, m=1, 2, . . . , l).

Node V₃ may calculate an average LLR as expressed by Equation 34.

$\begin{matrix}{{LLR}_{i} = {{L\; L\; {R\left( {X_{i,3}^{1}X_{i,3}^{2}\mspace{14mu} \ldots \mspace{14mu} X_{i,3}^{1}} \right)}} = \frac{\sum\limits_{m = 1}^{1}{L\; L\; {R\left( X_{i,3}^{m} \right)}}}{1}}} & \left\lbrack {{Equation}\mspace{14mu} 34} \right\rbrack\end{matrix}$

In this example, node V₃ may select thresholds (T₃ ¹ and T₃ ²) inadvance. T₃ ¹ and T₃ ² denote LLR qualities of data streams from node V₁and node V₂.

Based on the assumptions set forth herein, three cases may be possiblebased on a reliability of Y₃ ^(m) of node V₃.

-   -   Case 1: LLR1>T₃ ¹ and LLR2<T₃ ² or LLR1<T₃ ¹ and LLR2>T₃ ²    -   Case 2: all LLRi>T₃ ^(i) for i=1, 2    -   Case 3: all LLRi≦T₃ ^(i) for i=1, 2

In Case 3, node V₃ may not perform transmission in the same manner asthe BPSK case.

Case 1 describes an example in which LLR1>T31 and LLR2<T32 without lossof generality. In this example, node V₃ may generate a transmissionsignal ({tilde over (Y)}₃ ^(m)) (m=1, 2, . . . , l) that has p(X_(1,3)^(m)|{tilde over (Y)}₃ ^(m)) that has the same a posteriori probabilitydistribution as p(X_(1,3) ^(m)|Y₃ ^(m)), and that minimizes an expectedpower of error, as given in Equation 35. A method of generating {tildeover (Y)}₃ ^(m) has been described herein.

{tilde over (Y)} ₃ ^(m)=α_(1,3) X _(1,3) ^(m) +ñ ₃ ^(m)  [Equation 35]

In this example, {tilde over (Y)}₃ ^(m) may have the same LLR as LLR1with respect to X_(1,3m), and thus, from a point of view of receivernode t₁, {tilde over (Y)}₃ ^(m) may be equivalent to X_(1,3) ^(m) and{tilde over (Y)}₃ ^(m) may not carry information that is associated withX_(2,3) ^(m). Node V₃ may scale {tilde over (Y)}₃ ¹{tilde over (Y)}₃ ² .. . {tilde over (Y)}₃ ^(l) using a constant factor β so that a sequence(β{tilde over (Y)}₃ ^(m)) (m=1, 2, . . . , l) may have an average powerP. Node V₃ may perform scaling to satisfy another predetermined powerconstraint, such as a peak power. Node V₃ may transmit β{tilde over(Y)}₃ ^(m). In this example, all information that is associated withX_(2,3) ^(m) may be removed from a header of β{tilde over (Y)}₃ ^(m)That is, identification information that is associated with node V₁corresponding to X_(1,3) ^(m) may be included in the header of β{tildeover (Y)}₃ ^(m). Accordingly, information that is associated withtransmission nodes corresponding to symbols that have high reliabilitiesmay be included in a header of a sequence.

Case 2 describes an example in which all symbols have highreliabilities, respectively.

Here, a set of r₁r₂ possible values of Σ_(i=1) ²α_(i,3)X_(i,3) ^(m),given by A={α_(1,3)c₁+α_(2,3)c₂|c₁εA₁,c₂εA₂}, may be used. In thisexample, Y₃ ^(m) of node V₃ may be expressed by Equation 36.

Y ₃ ^(m) =X ₃ ^(m) +n ₃ ^(m) (m=1,2, . . . ,l)  [Equation 36]

Also, X₃ ^(m)=α_(1,3)X_(1,3) ^(m)+α_(2,3)X_(2,3) ^(m)εA. Node V₃ maycalculate {tilde over (Y)}₃ ^(m) that has p(X₃ ^(m)|{tilde over (Y)}₃^(m)) that has the same a posteriori distribution as p(X₃ ^(m)|Y₃ ^(m))based on the Kullback-Leibler distance, and that minimizes an expectedpower of error with respect to all m=1, 2, . . . , l.

{tilde over (Y)} ₃ ^(m) =X ₃ ^(m) +ñ ₃ ^(m)  [Equation 37]

Node V₃ may scale {tilde over (Y)}₃ ¹{tilde over (Y)}₃ ² . . . {tildeover (Y)}₃ ^(l) using a constant factor β so that β{tilde over (Y)}₃^(m) (m=1, 2, . . . , l) may have an average power P Also, the scalingmay be performed to satisfy another predetermined power constraint, suchas a peak power. Node V₃ may transmit β{tilde over (Y)}₃ ^(m). In thisexample, information that is associated with transmission nodescorresponding to symbols that have high reliabilities may be included ina sequence header. Accordingly, information that is associated with allthe nodes in Case 2 may be included in a header β{tilde over (Y)}₃ ^(m).

<Three or More Transmission Nodes>

The descriptions herein may be extended to a case in which three or moretransmission nodes corresponding to source nodes perform transmission toa single relay node. Assuming that K transmission nodes exist, a signalconstellation of a node V_(i)=S_(i) (i=1, 2, . . . , k) is referred toas A_(i) (i=1, 2, . . . , k). In this example, A_(i), includes r_(i)(i=1, 2, . . . , k) elements. A received signal of node V_(j) isexpressed by Equation 38.

$\begin{matrix}{Y_{j}^{m} = {{\sum\limits_{i = 1}^{k}{\alpha_{i,j}X_{i,j}^{m}}} + n_{j}^{m}}} & \left\lbrack {{Equation}\mspace{14mu} 38} \right\rbrack\end{matrix}$

(m=1, 2, . . . , l)

In Equation 38, α_(i,j) is a channel coefficient of a channel betweennode V_(i) (i=1, 2, . . . , k) and node V₃.

For each of i=1, 2, m=1, 2, . . . , l and cεAi, an LLR may be calculatedas expressed by Equation 39.

$\begin{matrix}{{{LLR}_{i,m}(c)} = {\ln \frac{p\left( {X_{i,j}^{m} = \left. c \middle| r \right.} \right)}{p\left( {X_{i,j}^{m} \neq c} \middle| r \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 39} \right\rbrack\end{matrix}$

Here,

${L\; L\; {R\left( X_{i,j}^{m} \right)}} = {\max\limits_{c \in _{i}}{{LLR}_{i,m}(c)}}$

(i=1, 2, . . . , k and m=1, 2, . . . , l)

Node V_(j) may calculate an average LLR as expressed by Equation 40.

$\begin{matrix}{{LLR}_{i} = {{L\; L\; {R\left( {X_{i,j}^{1}X_{i,j}^{2}\mspace{14mu} \ldots \mspace{14mu} X_{i,j}^{1}} \right)}} = \frac{\sum\limits_{m = 1}^{1}{L\; L\; {R\left( X_{i,j}^{m} \right)}}}{1}}} & \left\lbrack {{Equation}\mspace{14mu} 40} \right\rbrack\end{matrix}$

(for i=1, 2, . . . , k)

In this example, node V_(j) may select thresholds (T_(j) ^(i)) (i=1, 2,. . . , k), in advance. T_(j) ^(i) denotes LLR qualities of datastreams, appropriate for decoding, which are from node V_(i). Node V_(j)may select all of the nodes satisfying LLRi>T_(j) ^(i), that is, all ofthe nodes that have high reliabilities. If a node satisfying LLRi>T_(j)^(i) does not exist, node V_(j) may determine that none of symbols inthe stream X_(i,j) ¹X_(i,j) ² . . . X_(i,j) ^(l) (i=1, 2, k) have a highreliability and may not transmit a stream associated with X_(i,j)¹X_(i,j) ² . . . X_(i,j) ^(l). Any further amplification orre-transmission may decrease a quality of underlying stream in atransmission signal. Accordingly, node V₃ may cease transmission and mayconsume energy.

In this example, LLRi>(i=1, 2, . . . , k_(l)) and LLRi≦T_(j) ^(i)(k₁<i≦k), for k₁ satisfying 1≦k1≦k, without loss of generality. A=A₁×A₂×. . . ×A_(k) ₁ r₁, r₂, . . . r_(k) linear sums

${\left( {\alpha_{i,j},\ldots \mspace{14mu},\alpha_{k_{1,j}}} \right)} = \left\{ {\sum\limits_{i = 1}^{k_{1}}{\alpha_{i,j}a_{i}}} \middle| {a_{i} \in _{i}} \right\}$

are used. For each m=1, 2, . . . , l, node V_(j) may calculate Equation41.

$\begin{matrix}{{{{LLR}_{i,m}(c)} = {\ln \frac{p\left( {{\sum\limits_{i = 1}^{k_{1}}{\alpha_{i,j}X_{i,j}^{m}}} = \left. c \middle| r \right.} \right)}{p\left( {{\sum\limits_{i = 1}^{k_{1}}{\alpha_{i,j}X_{i,j}^{m}}} \neq c} \middle| r \right)}}}{c \in {\left( {\alpha_{1,j},\ldots \mspace{14mu},\alpha_{k_{1,j}}} \right)}}} & \left\lbrack {{Equation}\mspace{14mu} 41} \right\rbrack\end{matrix}$

Node V_(j) may generate a transmission signal ({tilde over (Y)}_(j)^(m)) that is modeled as given in Equation 42.

{tilde over (Y)} _(j) ^(m) =X+ñ  [Equation 42]

XεA is a signal that is transmitted from each transmission node. ñcomplex Gaussian noise that has a mean of zero and a variance of σ² percomplex dimension. Node V_(j) may generate {tilde over (Y)}_(j) ^(m)that has p(X_(j) ^(m)|{tilde over (Y)}_(j) ^(m)) the same a posteriorprobability distribution as p(X_(j) ^(m)|Y_(j) ^(m)) based on theKullback-Leibler distance, and that minimizes an expected power of errorwith respect to all m=1, 2, . . . , l.

Node V_(j) may scale {tilde over (Y)}_(j) ¹{tilde over (Y)}_(j) ² . . .{tilde over (Y)}_(j) ^(l) using a constant factor β so that a sequence(β{tilde over (Y)}_(j) ^(m)) (m=1, 2, . . . , l) may have an averagepower P. Also, the scaling may be performed to satisfy anotherpredetermined power constraint, such as a peak power. Node V_(j) maytransmit the scaled sequence to receiver nodes. For example, node V_(j)may transmit a scaled sequence to receiver nodes.

<Transmission from Arbitrary Nodes to Arbitrary Nodes>

A case in which node V_(i) (i=1, 2, . . . , k) is not necessarily asource node is described. A transmission signal transmitted from eachnode may have a structure including a linear sum of source signals andnoise. The structure is true for nodes in a layer 0 of a network, thatis, source nodes. It should also be appreciated that a transmissionsignal in a layer 1 may have the same structure.

In this example, it is inductively assumed that node V, (i=1, 2, . . . ,k) in a layer q (q>1) transmits a signal corresponding to a linear sumof source signals that are influenced by noise. A support of the signaltransmitted by node V_(i) is referred to as supp(v_(i)). In thisexample, supp(v_(i)) may be a set of all source nodes S₁, S₂, . . . ,S_(N) that appear in the linear summation part of the signal transmittedby node V_(i). Also, supp(v_(i), i=1, 2, . . . , k)=∪_(i=1)^(k)supp(V_(i)). A signal at node V_(j) may be a linear sum of signalsof source nodes appear in supp(v_(i), i=1, 2, . . . , k). In relation tonode V_(j), it is similar to a case in which sources nodes in ∪_(i=1)^(k)supp(v_(i)) directly perform transmission to node V_(j), withselected channel coefficients.

Accordingly, node V_(j) may apply the network coding method describedherein. Similarly, a signal transmitted from node V_(j) may also be asignal corresponding to a linear sum of signals, influenced by noise,which are transmitted from source nodes. Accordingly, it is inductivelyverified that an arbitrary node may transmit a signal corresponding to alinear sum of source signals influenced by noise. Therefore, thephysical layer network coding method may be applicable to an arbitrarynetwork such as another relay node.

<Network Coding Header>

An example of a header for the network coding method is described.According to the physical layer network coding method, a header of asignal transmitted by node V_(j) may include only informationsupp(v_(j)). For example, the header may include indices of only sourcenodes included in the support.

In this example, pilot sequences that an intermediate node is aware of,in advance, may be included in respective packets that are transmittedfrom source nodes so that an intermediate node corresponding to a relaynode or a receiver node corresponding to a destination node may estimateeffective channel coefficients. The pilot sequences may not be regardedas overhead because the pilot sequences may be used for channelestimation in a communication system that does not use a network codingscheme.

<Decoding in Destination Node>

An example of a reception process in each destination node is described.Based on a transmission strategy of a prior layer, each destination node(T_(j)) (j=1, 2, . . . , M) may receive M_(j) linear sums of sourcepackets that include noise, from N_(j)≦N source nodes S₁, . . . , S_(N).This process is similar to a wireless multi-user transmission thatincludes N, transmitters and M_(j) reception antennas. This process isalso similar to an N_(j)×M_(j) multiple-input multiple-output (MIMO)transmission scenario in which transmission antennas transmitindependent coded signals. Therefore, various MIMO receivers, forexample, a BLAST receiver, a full maximum likelihood receiver, and thelike, may be applied to T_(j) to decode source packets. For example, ifN_(j)=M_(j), a MIMO channel inversion may be applicable to separation ofsignals transmitted from various source nodes.

<Method of Enhancing Robustness and Scalability>

Each node may use a multilevel structure to determine a constellation toobtain more enhanced robustness and scalability. For example, a QPSKconstellation may be a scaled sum of a BPSK constellation, as given byEquation 43.

$\begin{matrix}{{Q\; P\; S\; K} = {{\frac{\sqrt{2}}{2}B\; P\; S\; K} + {\sqrt{- 1}\frac{\sqrt{2}}{2}B\; P\; S\; K}}} & \left\lbrack {{Equation}\mspace{14mu} 43} \right\rbrack\end{matrix}$

A 16-QAM constellation may be given by Equation 44.

$\begin{matrix}{{16\text{-}Q\; A\; M} = {{\frac{2}{\sqrt{5}}Q\; P\; S\; K} + {\sqrt{- 1}\frac{1}{\sqrt{5}}Q\; P\; S\; K}}} & \left\lbrack {{Equation}\mspace{14mu} 44} \right\rbrack\end{matrix}$

In the same manner, 32-QAM, and 64-QAM may be expressed by scaledversions of BPSK and QPSK constellations. For example, a source nodeS_(i) that performs transmission based on the 16-QAM constellation maybe a sum of virtual sources S_(i) ¹ and S_(i) ² that transmits QPSKsymbols at channel gains of

$\frac{2}{\sqrt{5}}\mspace{14mu} {and}{\mspace{11mu} \;}{\frac{1}{\sqrt{5}}.}$

Although one of the virtual sources transmits a signal that has a lowreliability, the robustness may be enhanced because the other virtualsource may transmit a signal.

FIG. 4 illustrates an example of a network to which a physical layernetwork coding method is applied.

Referring to FIG. 4, two source nodes V₀=S₁ and V₁=S₂ exist in a layer0, two relay nodes V₂ and V₃ exist in a layer 1, nodes V₄ and V₅ existin a layer 2, and destination nodes V₆=T₁ and V₇=T₂ exist in a layer 3.

A channel gain α_(i,j) between node V_(i) in each layer k (k=0, 1) andnode in a layer k+1 may be modeled as i.i.d. samples of a circularlysymmetric complex Gaussian N(0,1) that have a variance of 0.5 per realdimension, with respect to all i and j. In this example, a receivedsignal of node V_(j) is corrupted by i.i.d. samples of a circularlysymmetric complex Gaussian N(0, σ_(j) ²) that have a variance of σ_(j)²/2 per real dimension. Also, an average transmission power for eachtransmission node is assumed to be 1. Accordingly, an SNR may be definedas

$1/{{\sigma_{2}^{2}}^{\frac{1}{\sigma_{2}2}}.}$

The transmission from V₀=S₁ and V₁=S₂ may be simultaneously received bynode V₂ and node V₃ which correspond to relay nodes, and transmissionfrom node V₂ and node V₃ may be simultaneously received by node V₄ andnode V₅ which correspond to relay nodes. Transmission from node N₄ andtransmission from node V₅ may be performed at different times, and maybe separately received by nodes V₆=T₁ and V₇=T₂ corresponding todestination nodes. The source nodes may perform transmission using anuncoded QPSK. For simplicity, all thresholds for reliability are assumedto be zero. Each receiver may perform maximum likelihood (ML) decoding.Hereinafter, results of simulation based on scenarios described hereinare provided with reference to FIGS. 5 through 7.

FIGS. 5 through 7 illustrate examples of a result of a simulation basedon a physical layer network coding method that uses reliability.

FIGS. 5 and 6 illustrate a comparison between a result of a simulationwith respect to

σ₂² = σ₃² = 2σ₄² = 2σ₅² = 2σ₆² = 2σ₇²

and a result based on a conventional analog-and-forward scheme. Thephysical layer network coding method that uses reliability may enhanceperformance by about 2 dB when compared to the conventional scheme. Ascan be seen, the SNR is improved in the physical network coding methodthat uses reliability.

FIG. 7 shows a result of a simulation with respect to

2σ₂² = 2σ₃² = σ₄² = σ₅² = σ₆² = σ₇².

In this example, a gain may be enhanced by about 1 dB in FIG. 7 becausean amount of noise to be controlled at a first stage is relativelysmall, unlike FIGS. 5 and 6.

The example simulation results from FIGS. 5 through 7 illustrate thatthe physical layer network coding method may enhance robustness andscalability in comparison to the conventional method. More enhancedsimulation result may be obtained based on a network structure or achannel environment.

FIG. 8 illustrates an example a communication method of a relay nodebased on a physical layer network coding method that uses reliability.

Referring to FIG. 8, the relay node receives a signal that includes afirst symbol through k^(th) symbols that are transmitted from a firstnode through a k^(th) node, that is, from transmission nodes in 810. Forexample, the relay node may estimate, in advance, channels with respectto the first node through the k^(th) node respectively, based on pilotstransmitted from the first node through the k^(th) node.

The relay node respectively calculates the reliabilities of the firstsymbol through the k^(th) symbol based on a predetermined criterion, in820. For example, the relay node may calculate reliabilities of thefirst symbol through the k^(th) symbol, based on corresponding LLRs ofthe first symbol through the k^(th) symbol. Each LLR may be calculatedbased on the received signal and corresponding channel information thatis associated with the first node through the k^(th) node.

The relay node selects symbols that have reliabilities that are greaterthan or equal to the predetermined criterion from the first symbolthrough the k^(th) symbol, in 830.

The relay node generates a transmission signal that maintains thereliabilities of selected symbols and that excludes components that areassociated with symbols that have reliabilities that are less than thepredetermined criterion, in 840. In this example, the relay node maygenerate the transmission signal to decrease an expected power of errorbetween the transmission signal of the relay node and a signalcorresponding to the received signal excluding noise. For example, therelay node may generate the transmission signal of which LLRs of theselected symbols that are equivalent to LLRs of the selected symbols inthe received signal or that are different in a predetermined range. Asan example, the equivalency or the difference may be determined based onthe Kullback-Leibler distance, or may be based on a different standard.

In 850, a relay node transmits the transmission signal to receivernodes, that is, a first destination node through a k^(th) destinationnode that correspond to the first transmission node through the k^(th)transmission node, in 850. The relay node may broadcast the transmissionsignal. The relay node may scale the transmission signal based on apredetermined transmission power and may transmit the scaledtransmission signal. The relay node may transmit identificationinformation that is associated with nodes corresponding to the selectedsymbols, that is, symbols that have high reliabilities.

FIG. 9 illustrates an example of a relay node that uses a physical layernetwork coding method that uses reliability.

Referring to FIG. 9, the relay node includes a receiving unit 910, aprocessing unit 920, and a transmitting unit 930.

The receiving unit 910 may receive a signal that includes a first symbolthrough a k^(th) symbol that is transmitted from a first node through ak^(th) node.

The processing unit 920 may calculate the reliabilities of the firstsymbol through the k^(th) symbol, respectively, based on a predeterminedcriterion. The processing unit 920 may select symbols that havereliabilities that are greater than or equal to the predeterminedcriterion from the first symbol through the k^(th) symbol. Theprocessing unit 920 may generate, based on the received signal, atransmission signal that maintains the reliabilities of the selectedsymbols and that excludes components that are associated with symbolsthat have reliabilities that are less than the predetermined criterion.

The transmitting unit 930 may transmit the transmission signal.

A relay node and a communication method of the relay node have beenpreviously described. That is, the examples described herein withreference to FIGS. 1 through 7 are also applicable to the relay node ofFIG. 9 and the communication method of the relay node of FIG. 8, andthus, further descriptions thereof is omitted for conciseness.

Various examples are directed towards a relay node that determines thereliability of symbols corresponding to a plurality of nodes, based on asignal received from the plurality of nodes. The relay node may generatea transmission signal to maintain the reliability of symbols having highreliabilities and to exclude components of symbols that have lowreliabilities. Accordingly, the relay node may reduce an amount of powerconsumed for transmitting symbols that have low reliabilities and mayconsume a greater amount of power for transmitting symbols that havehigh reliabilities. Accordingly, transmission efficiency may increase.

In various examples, the relay node may generate a transmission signalthat reduces an expected power of error based on a signal that isreceived from a plurality of nodes, and thus, may reduce the effect ofnoise.

In various examples, a transmission signal generating method of a relaynode may have a high scalability because the method may be performedregardless of a change in a number of nodes.

Program instructions to perform a method described herein, or one ormore operations thereof, may be recorded, stored, or fixed in one ormore computer-readable storage media. The program instructions may beimplemented by a computer. For example, the computer may cause aprocessor to execute the program instructions. The media may include,alone or in combination with the program instructions, data files, datastructures, and the like. Examples of computer-readable storage mediainclude magnetic media, such as hard disks, floppy disks, and magnetictape; optical media such as CD ROM disks and DVDs; magneto-opticalmedia, such as optical disks; and hardware devices that are speciallyconfigured to store and perform program instructions, such as read-onlymemory (ROM), random access memory (RAM), flash memory, and the like.Examples of program instructions include machine code, such as producedby a compiler, and files containing higher level code that may beexecuted by the computer using an interpreter. The program instructions,that is, software, may be distributed over network coupled computersystems so that the software is stored and executed in a distributedfashion. For example, the software and data may be stored by one or morecomputer readable storage mediums. Also, functional programs, codes, andcode segments for accomplishing the example embodiments disclosed hereincan be easily construed by programmers skilled in the art to which theembodiments pertain based on and using the flow diagrams and blockdiagrams of the figures and their corresponding descriptions as providedherein. Also, the described unit to perform an operation or a method maybe hardware, software, or some combination of hardware and software. Forexample, the unit may be a software package running on a computer or thecomputer on which that software is running.

As a non-exhaustive illustration only, a terminal/node described hereinmay refer to mobile devices such as a cellular phone, a personal digitalassistant (PDA), a digital camera, a portable game console, and an MP3player, a portable/personal multimedia player (PMP), a handheld e-book,a portable lab-top PC, a global positioning system (GPS) navigation, atablet, a sensor, and devices such as a desktop PC, a high definitiontelevision (HDTV), an optical disc player, a setup box, a homeappliance, and the like that are capable of wireless communication ornetwork communication consistent with that which is disclosed herein.

A computing system or a computer may include a microprocessor that iselectrically connected with a bus, a user interface, and a memorycontroller. It may further include a flash memory device. The flashmemory device may store N-bit data via the memory controller. The N-bitdata is processed or will be processed by the microprocessor and N maybe 1 or an integer greater than 1. Where the computing system orcomputer is a mobile apparatus, a battery may be additionally providedto supply operation voltage of the computing system or computer. It willbe apparent to those of ordinary skill in the art that the computingsystem or computer may further include an application chipset, a cameraimage processor (CIS), a mobile Dynamic Random Access Memory (DRAM), andthe like. The memory controller and the flash memory device mayconstitute a solid state drive/disk (SSD) that uses a non-volatilememory to store data.

A number of examples have been described above. Nevertheless, it shouldbe understood that various modifications may be made. For example,suitable results may be achieved if the described techniques areperformed in a different order and/or if components in a describedsystem, architecture, device, or circuit are combined in a differentmanner and/or replaced or supplemented by other components or theirequivalents. Accordingly, other implementations are within the scope ofthe following claims.

1. A communication method of a relay node, the method comprising:receiving a signal including a first symbol through a k^(th) symbol thatare transmitted from a first node through a k^(th) node, respectively;calculating, based on a predetermined criterion, a reliability of thefirst symbol through the k^(th) symbol, respectively; selecting one ormore symbols from among the first through the k^(th) symbol that have areliability that is greater than or equal to the predeterminedcriterion; and generating a transmission signal that maintains thereliabilities of the selected symbols and that excludes components ofsymbols that have reliabilities which are less than the predeterminedcriterion.
 2. The method of claim 1, wherein the generating comprises:generating the transmission signal to decrease an expected power oferror between the transmission signal and the received signal.
 3. Themethod of claim 1, wherein the calculating comprises: calculating thereliabilities of the first symbol through the k^(th) symbol, based on alog likelihood ratio (LLR) with respect to the first symbol through thek^(th) symbol, respectively, each LLR calculated based on the receivedsignal and channel information that is associated with the first nodethrough the k^(th) node, respectively.
 4. The method of claim 1, whereinthe generating comprises: generating the transmission signal in whichLLRs of the selected symbols are equivalent to LLRs of the selectedsymbols in the received signal or are within a predetermined range. 5.The method of claim 4, wherein the equivalence or the difference in thepredetermined range is determined based on the Kullback-Leiblerdistance.
 6. The method of claim 1, further comprising: estimatingchannels with respect to the first node through the k^(th) node based onpilots that are transmitted from the first node through the k^(th) node,respectively.
 7. The method of claim 1, further comprising: transmittingthe transmission signal to a first destination node through a k^(th)destination node corresponding to the first node through the k^(th)node, respectively.
 8. The method of claim 7, wherein the transmittingcomprises: transmitting the transmission signal by scaling thetransmission signal based on a predetermined transmission power.
 9. Themethod of claim 1, further comprising: transmitting identificationinformation that is associated with nodes corresponding to the selectedsymbols.
 10. A computer-readable storage medium having stored thereinprogram instructions to cause a processor to implement a communicationmethod of a relay node, the method comprising: receiving a signalincluding a first symbol through a k^(th) symbol that are transmittedfrom a first node through a k^(th) node, respectively; calculating,based on a predetermined criterion, a reliability of the first symbolthrough the k^(th) symbol, respectively; selecting one or more symbolsfrom among the first through the k^(th) symbol that have a reliabilitythat is greater than or equal to the predetermined criterion; andgenerating a transmission signal that maintains the reliabilities of theselected symbols and that excludes components of symbols that havereliabilities which are less than the predetermined criterion.
 11. Arelay node comprising: a receiving unit to receive a signal including afirst symbol through a k^(th) symbol that are transmitted from a firstnode through a k^(th) node, respectively; and a processing unit tocalculate, based on a predetermined criterion, reliabilities of thefirst symbol through the k^(th) symbol, respectively, wherein theprocessing unit selects one or more symbols that have a reliability thatis greater than or equal to the predetermined criterion from among thefirst symbol through the k^(th) symbol, and generates a transmissionsignal that maintains the reliabilities of the selected symbols and thatexcludes components of symbols that have reliabilities which are lessthan the predetermined criterion.
 12. The relay node of claim 11,wherein the processing unit generates the transmission signal todecrease an expected power of error between the transmission signal andthe received signal.
 13. The relay node of claim 11, wherein theprocessing unit calculates the reliabilities of the first symbol throughthe k^(th) symbol, based on log likelihood ratio (LLR) with respect tothe first symbol through the k^(th) symbol, respectively, each LLRcalculated based on the received signal and channel information that isassociated with the first node through the k^(th) node, respectively.14. The relay node of claim 11, wherein the processing unit generatesthe transmission signal in which LLRs of the selected symbols areequivalent to LLRs of the selected symbols in the received signal or arewithin a predetermined range.
 15. The relay node of claim 14, whereinthe equivalence or the difference in the predetermined range isdetermined based on the Kullback-Leibler distance.
 16. The relay node ofclaim 11, wherein the processing unit estimates channels with respect tothe first node through the k^(th) node based on pilots that aretransmitted from the first node through the k^(th) node, respectively.17. The relay node of claim 11, further comprising: a transmitting unitto transmit the transmission signal to a first destination node througha k^(th) destination node corresponding to the first node through thek^(th) node, respectively.
 18. The relay node of claim 17, wherein thetransmitting unit scales the transmission signal based on apredetermined transmission power, and transmits the scaled transmissionsignal.
 19. The relay node of claim 11, wherein the transmitting unittransmits identification information that is associated with nodescorresponding to the selected symbols.
 20. A relay node in a wirelessnetwork, the relay node comprising: a receiver configured to receivesymbols from one or more nodes that are within the wireless network; aprocessor configured to determine a reliability of each received symbolbased on a physical layer network coding method that uses reliability;and a transmitter configured to transmit only those received symbolsthat are determined to have a reliability above a threshold.
 21. Therelay node of claim 20, wherein the physical layer network coding methodcalculates the reliability of a received symbol based on a loglikelihood ratio (LLR) of the received symbol and channel informationthat is associated with a node that transmitted the received symbol. 22.The relay node of claim 20, wherein the receiver is further configuredto simultaneously receive a first symbol from a first node and a secondsymbol from a second node, and the processor is further configured todetermine a reliability of the first symbol and the second symbol basedon the physical layer network coding method that uses reliability. 23.The relay node of claim 22, wherein, in response to the receiverdetermining that the first symbol has a reliability above the threshold,and that the second symbol has a reliability below the threshold, thetransmitter is further configured to transmit a transmission signalincluding the first symbol and excluding the second symbol.